<h2>Problem 202</h2>
<div style="color:#666;font-size:80%;">05 July 2008</div><br />
<div class="problem_content">
<p>Three mirrors are arranged in the shape of an equilateral triangle, with their reflective surfaces pointing inwards. There is an infinitesimal gap at each vertex of the triangle through which a laser beam may pass.</p>

<p>Label the vertices A, B and C. There are 2 ways in which a laser beam may enter vertex C, bounce off 11 surfaces, then exit through the same vertex: one way is shown below; the other is the reverse of that.</p>

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  <img src="project/images/p_201_laserbeam.gif" alt="" />
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<p>There are 80840 ways in which a laser beam may enter vertex C, bounce off 1000001 surfaces, then exit through the same vertex.</p>

<p>In how many ways can a laser beam enter at vertex C, bounce off 12017639147 surfaces, then exit through the same vertex?</p>
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